2014-2015 Schedule

Giving a Talk

The graduate seminar is held approximately once a month, typically on the first Thursday. The general format is as follows: pizza at 4:30, the talk at 5:00, and we go to the bar afterwards.

If you'd like to give a talk, please email Matt Sourisseau (sourisse [at] math.utoronto.ca) or Jerrod Smith (jerrod.smith [at] utoronto.ca) with your talk title and what day(s) you would prefer. We will then ask you for a short (1-2 paragraph) abstract the week before the talk so that it can go in the official seminar announcements and posters.

Archive

Not to be confused with certain small birds or equestrian harnesses, a martingale is a concept in probability with interesting applications to PDEs, functional analysis, and mathematical finance. We'll begin by motivating the idea of a martingale as a fair game, and from there give a neat characterization of a zany process called Brownian motion with an intimate connection to the heat equation. We'll then discuss some results concerning martingales in the areas of Hardy spaces and asset pricing.

The study of knots in three-dimensional space is often done through their projections, called "knot diagrams", onto different planes. A central question then is how to tell, given two such diagrams, whether they represent the same knot. This is typically answered through functions called "knot invariants", the most classical among which is the Alexander polynomial. We will show three different ways of constructing it and mention some of its nice properties. Named after its creator, James Alexander, it first appeared in a paper in 1928 and we will discuss its relevance today, including generalizations to tangles and, time-permitting, categorification.

Cluster algebras are commutative algebras having distinguished "clusters" of generators, with different clusters being related to each other by "mutations". Despite the terminology they are relatively elementary to define (and do not require extensive background), and come equipped with a wealth of nice properties.

Since their definition by Fomin and Zelevinsky (2002), cluster algebras have found numerous applications in geometry, representation theory, physics, and elsewhere. We will provide a brief tour of this theory and some of its many applications.

Given the recent excitement in mathematical circles over G. Perelman’s solution to the Poincaré Conjecture, you have most likely heard of the terms "Ricci flow". But what exactly is Ricci flow, you may be wondering? The aim of my talk will be to provide an answer to this question. I will briefly cover some of the historical context in which Ricci flow was developed. I then hope to provide an intuitive understanding of what Ricci flow is by discussing a few of its interesting properties and solutions. Lastly, I will comment on some of the resent applications of Ricci flow to current research.

Tracey BalehowskyUniversity of Toronto

January 30 @ 6PM

Here Be Dragons: An Introduction to the Independence Phenomenon (abstract)

Everything you wanted to know about Snakes & Ladders but were too young to ask (abstract)

The traditional children's game, Snakes & Ladders, is currently sold as Chutes
& Ladders by Hasbro. An e-tailing description of the game advertises:

"Chutes and Ladders is ideal for younger children who are still learning to take
turns and just beginning to recognize numbers (the spinner stays in the single digits). It's also a gentle introduction to the higher numbers as players climb to 100 at the top of the board. And, thanks to all those chutes and ladders, it's got enough excitement to keep your 7-year-old on the edge of her seat."

Linear algebra suggests otherwise. Take a chance, hear the talk, but decide for
yourself if this "Family Game Night game for preschoolers" really belongs on your holiday shopping list.

I will introduce one of the major mathematical models of 'topological quantum computation', the Kitaev's quantum double model. As will be argued by Alexander Kirillov on Thursday, this family of models defined on planar graphs is related to another family, called the Levin-Wen models. Both families can be viewd as extended 3-2-1(-0) TQFTs (top dimension is 3 because of 2 spatial + 1 temporal dimensions) arising from the Turaev-Viro construction on the tensor category labeling the underlying 'string-nets'; alternatively, the same extended TQFTs may also be obtained by applying the Reshetikhin-Turaev construction on the tensor category of 'excitations'. Without further ado, I will dive into the Kiteav's description of these models, where the representation theory of quantum doubles (of certain Hopf algebras) plays an important role. A cool thing about these models for topological quantum computation is that (if implemented,) they provide physical realizations of recently invented/popularized mathematical notions of extended TQFTs and higher categories. We can even apply the trendy concept of categorification in this context and produce extended TQFTs for higher dimensional manifolds (e.g. a 4-3-2-1-0 theory might describe our space-time manifold).

The Kardar-Parisi-Zhang equation has been widely used in physics as a model of random growth and directed polymers. We will discuss the sense that can be made of the equation, and the non-Gaussian fluctuations that describe its universality class.

In this talk I plan to discuss one of my recent research interests, which involves the application of mathematical techniques from the theory of optimal transportation to one of the central problems in economic theory: Optimal pricing facing informational asymmetry. The monopolist's problem of deciding what types of products to manufacture and how much to charge for each of them, knowing only statistical information about the preferences of an anonymous field of potential buyers, is one of the basic problems analyzed in economic theory. The solution to this problem when the space of products and of buyers can each be parameterized by a single variable (say quality X, and income Y) garnered Mirrlees (1971) and Spence (1974) their Nobel prizes in 1996 and 2001. The multidimensional version of this question is a largely open problem in the calculus of variations (see Basov's book "Multidimensional Screening".) I plan to describe recent work with A Figalli and Y-H Kim, identifying structural conditions on the value b(X,Y) of product X to buyer Y which reduce this problem to a convex program in a Banach space - leading to uniqueness and stability results for its solution, confirming robustness of certain economic phenomena observed by Armstrong (1996) such as the desirability for the monopolist to raise prices enough to drive a positive fraction of buyers out of the market, and yielding conjectures about the robustness of other phenomena observed Rochet and Chone (1998), such as the clumping together of products marketed into subsets of various dimension. The passage to several dimensions relies on ideas from differential geometry / general relativity, optimal transportation, and nonl
inear PDE.

Several years ago Braverman and Yampolsky showed that there exist parameters c such that the polynomial z2+c has a non-computable Julia set. However, this phenomenon is rare. For allmost all c, J_c is computable set, and we conjecture that it is computable in polynomial time. In my talk I will describe the proof of poly-time computability for a large class of parameters c, which is a natural step towards a proof of the conjecture.

We will show that Newton's 2nd law F = ma, which is 2nd order in time, can be naturally rewritten as a first order system (Hamilton's equations). This will lead us to define Poisson and symplectic manifolds, and we will see that many basic notions and results from classical mechanical systems can be interpreted geometrically. Armed with geometry, we will then examine some examples in detail, and (time permitting) say a few words about how symplectic geometry can be useful in areas of mathematics outside of classical mechanics.

This talk will give an introduction to some simple financial derivatives and the concepts used to price them. We will begin with a description of some basic equity derivatives and perform pricing by no-arbitrage arguments in a simple binomial model. We will then consider the continuous case where we derive and solve the Black-Scholes PDE using methods of Ito's Calculus. The talk will conclude with descriptions of more complicated equity derivatives and how the Black-Scholes PDE must be changed to represent the proper derivative value.

Leavitt path algebras are a natural generalization of the Leavitt algebras, which are a class of algebras introduced by Leavitt in 1962. For a directed graph $E$, the Leavitt path algebra $L_K(E)$ of $E$ with coefficients in $K$ has received much recent attention both from algebraists and analysts over the last decade. So far, some of the algebraic properties of Leavitt path algebras have been investigated, including primitivity, simplicity and being Noetherian.

First, we explicitly describe the generators of two-sided ideals in Leavitt path algebras associated to arbitrary graphs. We show that any two-sided ideal $I$ of a Leavitt path algebra associated to an arbitrary graph is generated by elements of the form $(v + \sum_{i=1}^n\lambda_i g^i)(v - \sum_{e\in S} ee^*)$, where $g$ is a cycle based at vertex $v$, and $S$ is a finite subset of $s^{-1}(v)$. Then, we use this result to describe the necessary and sufficient conditions on the arbitrary sized graph $E$, such that the Leavitt path algebra associated to $E$ satisfies two-sided chain conditions.

Convergence and smoothing properties of Steiner symmetrizations
and other simple rearrangements
(abstract)

Rearrangements were invented in the 19th century as tools for proving the isoperimetric inequality. In the 1970's, rearrangements gained new interest, because they proved useful for less obvious minimization problems in functional analysis.

Rearrangements manipulate the shape of a body (or a function) without changing its size (norm), usually by pushing it closer to a centered ball (or a radially decreasing function). For example, Steiner symmetrization introduces a hyperplane of symmetry, Schwarz symmetrization creates an axis of rotation, and polarization moves mass towards the origin across a hyperplane.

How well is full radial symmetrization approximated by sequences of such simple rearrangements? How fast do sequences converge, and what smoothing properties do they have? These key questions are subtle, because rearrangements are neither linear, nor spatially localized, and typically do not commmute. In this talk, I will discuss joint work with recent UofT Master's student Marc Fortier on these questions in the context of classical and recent results from the literature. Specifically, I will address conditions for (non)-convergence of sequences of Steiner symmetrizations and polarizations, and explain what happens for random sequences.

One basic result of Calculus 1 can be stated as follows. Suppose the Taylor expansion of a (sufciently smooth) function f(x) near the origin begins as f = x^p + ... + x^(p+1) + ... (with some numerical coeffcients). Then in some new (local) coordinates the function is f = x^p. The moral in higher dimension is: 'for a given power series near the origin only a very few monomials are important'. To formulate the precise criteria one uses the Newton diagram associated to f(x1,...,xn). (This helps to study degenerate minima/maxima, types of saddle points etc.) Today the Newton diagram is an important tool in various areas, e.g. understanding the local embedded geometry/topology of a hypersurface. In particular, if the germ of hypersurface is "generic enough" then the local topology is completely determined by its Newton diagrams. So, various hard topological questions are translated into combinatorial ones. Such hypersurface singularities (non-degenerate with respect to their diagrams) are easy to work with and serve as "baby models". I will give a short introduction to this topic.

Consider a local complex plane analytic curve near the origin. By implicit function theorem a smooth curve can be locally rectiË‰ed, i.e. the curve becomes a line after a local change of coordinates. Singular curves have more local structure. I will speak about the topological and analytic classifations of plane curve singularities. The former is discrete and has simple complete invariants. The latter has moduli. Then I will speak about various invariants (Milnor fibre and its cohomology lattice, links and their embeddings etc.)

It is a strange but important fact that there exist "sparse" combinatorial structures which, no matter how they are realized in Euclidean space, must exhibit a significant degrees of "complexity." Constructing such structures relies both on topological techniques (such as the Borsuk-Ulam theorem) and probabilistic techniques (such as the Chernoff bounds). These structures has provided many important counterexamples in geometry, such as counterexamples to the Baum-Connes conjecture. In the early '80's, two undergraduates proved that given any arrangement of n points in the plane, there must always exist a point in the plane which lies in at least two ninths of all the induced triangles. This theorem has some important consequences. Recently, it has been shown that there are sparse structures which get arbitrarily close to this bound.

Hermann Weyl was the first who studied seriously asymptotic distribution of eigenvalues (1911) of partial differential operators and formulated what waslater called Weyl's conjecture. In the first part of my talk I will discuss the history of the subject and ideas leading to the proof. This proof was worked out in the framework of Microlocal Analysis which is based on the idea of quantization, and the most useful quantization method was also suggested by H.Weyl (1927): http://en.wikipedia.org/wiki/Weyl_quantization. In the second part of the talk I will discuss quantization, uncertainty principle, quantum and classical dynamics. In the third part of the talk I discuss the proof: main singularity, the role of periodic trajectories in the remainder estimate and their possible contribution to the main part. I will follow the same presentation delivered few years ago: http://weyl.math.toronto.edu:8888/victor2/preprints/Grad_Talk_3.php.

In its simplest form, Hilbert's 5th problem asks the following: given a group that is a continuous manifold can we define a differentiable (or even better . analytic) structure on it? Gleason and Montgomery. Zippin in 1940s showed that something much stronger holds: any locally compact topological group "with no small subgroups" is automatically an analytic Lie group. We.ll prove this result in compact case and discuss the general case. Time permitting, we.ll discuss how Gromov used this result to prove his theorem that groups of polynomial growth are virtually nilpotent.

We are going to introduce covering arrays with row limit (CARLs) and
equivalent families of objects, such as graph coverings and group divisible
covering designs. Also, we are going to present some of the constructions
for CARLs with weight four. This is a talk of ongoing research in
combinatorics, but no previous knowledge of the subject is assumed.

We will look at some of the statements that are equivalent to the axiom of choice, like Zorn's Lemma and the Well-ordering Principle. As these statements are used throughout mathematics, every student should see their proofs at least once. We will also look at some "good" and "bad" consequences of the axiom of choice. Finally, we will look at a very bizarre equivalence to the axiom of choice.